Republicans both inside and outside the government warn that firing Mr. Mueller would be very unwise.
“You need to leave him alone, let him do his job and get it over with,” said Saxby Chambliss, a retired senator from Georgia who was the top Republican on the Intelligence Committee and talks frequently with his former colleagues about Mr. Trump and the special counsel. “The president keeps slamming Mueller, and I think that politically just doesn’t play well, even with folks I know who are part of the president’s base.”
Mr. McConnell’s argument has consistently been that such legislation is not needed because Mr. Trump will not take action against Mr. Mueller. But the president’s increasing anger and frustration about the inquiry has other Republicans joining Democrats in worrying that the counsel might be about to lose his job. Although top Republicans have said there would be significant blowback, it remains unclear what steps would be taken beyond admonishment and hand-wringing.
One top Senate Republican acknowledged privately that there may be little else the party would do. Any move toward impeachment seems out of the question in the Republican-controlled Congress. Conservatives in the House are instead demanding an end to the Mueller inquiry.
While the Senate could censure the president, it is doubtful Republicans would take that action before midterm elections in which they will be relying on Trump supporters to help them hold the Senate. It has been done only once before, in the case of Andrew Jackson (coincidentally a Trump favorite), and was later reversed.
Another Republican suggested that the Senate could refuse to consider Justice Department nominees unless the president agreed to the naming of a new counsel, a reaction Democrats would consider woefully insufficient. Democrats fear they might not learn the Republican reaction until after the president takes his own action.
“Why not head off a constitutional crisis at the pass, rather than waiting until it’s too late?” Senator Chuck Schumer of New York, the Democratic leader, said Thursday in calling for approval of the special counsel legislation. “Why even flirt with the prospect of a president challenging the very nature of our system of government?”
Democrats are just as concerned that Mr. Trump and House Republicans are building a separate case for firing Rod J. Rosenstein, the deputy attorney general, who is overseeing the investigation as a result of the decision by Attorney General Jeff Sessions to recuse himself. They fear that a move against Mr. Rosenstein would not grab as much attention as dismissing Mr. Mueller and that Mr. Rosenstein’s successor could then constrain the inquiry.
Pressed again Wednesday evening at a news conference about whether he intended to fire either of them, Mr. Trump repeated his denunciation of the idea of collusion between Russia and his campaign as a hoax and called for the inquiry to be wrapped up. “As far as the two gentlemen you told me about, they’ve been saying I’m going to get rid of them for the last three months, four months, five months,” he said. “And they’re still here.”
Despite Mr. McConnell’s resistance, Senator Charles E. Grassley, the Iowa Republican who is chairman of the Judiciary Committee, still plans for his panel to take up the special counsel bill next week. It appears to have a strong chance of receiving the committee’s approval.
“The views of the majority leader are important to consider, but they do not govern what happens here on the Judiciary Committee,” Mr. Grassley said Thursday.
Mr. McConnell has previously demonstrated that he is willing to bottle up a bill he believes could divide Senate Republicans or prove politically problematic, even if it has bipartisan support from members of the Judiciary Committee. In 2016, he refused to take up a criminal justice overhaul that could have passed the Senate with the backing of Mr. Grassley and many other Republicans, citing objections from a handful of conservatives.
Mr. Tillis and his allies say they still hope to sway Mr. McConnell. “It’s on us to convince the leader that it’s a worthwhile effort,” Mr. Tillis said. “We’ve got to get the votes, and that will come if we get a successful outcome out of Judiciary.”
The fight over the measure is taking place as James B. Comey, the former F.B.I. director, continues a high-profile media tour promoting his tell-all about clashes with Mr. Trump. Ms. Collins and other Republicans say they have found the entire exercise unseemly and damaging to the F.B.I. They contrasted Mr. Comey’s performance with what they say is the professionalism being exhibited by Mr. Mueller, a man well known to many lawmakers from his time at the F.B.I. and Justice Department.
“James Comey is no Bob Mueller,” Ms. Collins said.
Now she and many others are eager to protect Mr. Mueller from suffering Mr. Comey’s fate — fired by Mr. Trump over the Russia investigation.